Question 18
(USA 2) Let $[x]$ denote the greatest integer less than or equal to $x$. Pick any $x_1$ in $[0,1)$ and define the sequence $x_1,x_2,x_3,\ldots$ by $x_{n+1}=0$ if $x_n=0$ and $x_{n+1}=1/x_n-[1/x_n]$ otherwise. Prove that
$$x_1+x_2+\cdots+x_n\lt \frac{F_1}{F_2}+\frac{F_2}{F_3}+\cdots+\frac{F_n}{F_{n+1}},$$
where $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge1$.
$$x_1+x_2+\cdots+x_n\lt \frac{F_1}{F_2}+\frac{F_2}{F_3}+\cdots+\frac{F_n}{F_{n+1}},$$
where $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge1$.